Problem 39 - Algebra and Trigonometry with Analytic Geometry

Question

The temperature \(T\) within a cloud at height \(h\) (in feet) above the cloud base can be approximated using the equation \(T=B-\left(\frac{3}{1000}\right) h\), where \(B\) is the temperature of the cloud at its base. Determine the temperature at 10,000 feet in a cloud with a base temperature of \(55^{\circ} \mathrm{F}\) and a base height of 4000 feet. Note: For an interesting application involving the three preceding exercises, see Exercise 6 in the Discussion Exercises at the end of the chapter.

Step-by-Step Solution

Answer

The temperature at 10,000 feet is 37°F.

Step 1: Identify Given Variables and Equation

The given equation to find the temperature at a specific height is \( T = B - \left(\frac{3}{1000}\right) h \). Here, \( B = 55^{\circ} \mathrm{F} \) and \( h \) is the height above the cloud base in feet.

Step 2: Calculate Height Above Cloud Base

The cloud base height is given as 4000 feet, and we need the temperature at 10,000 feet. The height above the cloud base \( h \) is calculated as \( 10,000 - 4000 = 6000 \) feet.

Step 3: Apply the Equation to Find Temperature

Substitute \( B = 55^{\circ} \mathrm{F} \) and \( h = 6000 \) feet into the equation. \[ T = 55 - \left(\frac{3}{1000}\right) \times 6000 \]

Step 4: Perform the Calculation

Calculate the change in temperature: \( \left(\frac{3}{1000}\right) \times 6000 = 18 \).Subtract this from the base temperature: \( T = 55 - 18 = 37 \).

Step 5: Conclusion

The temperature at 10,000 feet within the cloud is 37°F.

Key Concepts

Understanding Temperature Calculation

When calculating temperature changes within a cloud, algebraic equations are a helpful tool. Temperature can vary with altitude, and in this problem, it is modeled by the equation \[ T = B - \left(\frac{3}{1000}\right)h \]Here, \(T\) represents the temperature at a specific height, \(B\) stands for the temperature at the cloud's base, and \(h\) is the height above that base. This equation reflects a lapse rate of \(3^{\circ}F\) per 1,000 feet, a common approximation used in meteorology. The negative sign indicates that temperature decreases as you ascend. Breaking down these calculations step-by-step makes it easier to determine temperature values at various heights. In practical use, you'd determine \(h\) by subtracting the cloud base height from the total height, then perform the multiplication and subtraction accordingly.

Knowing Cloud Temperature

Cloud temperature is crucial to understanding weather phenomena. By knowing the base temperature of a cloud, you can predict how it changes with altitude. For example, in the given problem, the cloud's base temperature \(B\) is \(55^{\circ}\mathrm{F}\).

• This means that the temperature at the cloud's formation (its lowest part) is \(55^{\circ}\mathrm{F}\).
• Understanding the base temperature helps to compute temperature at different heights within the cloud.

Using the equation, the cloud's temperature can be traced as it decreases with height, which can affect condensation and precipitation conditions. For instance, by substituting \(B\) and calculating the value of \(h\), you can find the specific cloud temperature at any given height, such as 10,000 feet in this exercise.

Importance of Height Above Base

Height above base is a key variable in determining temperature changes within clouds. In this exercise, the concept is straightforward: subtract the cloud base's height from the specific height where the temperature is needed.- Given the base height as 4,000 feet and a target height of 10,000 feet, the calculation is performed as follows: \( h = 10,000 - 4,000 = 6,000 \) feet.- This result, \( h = 6,000 \), is then plugged into the temperature equation.This measurement indicates how much the temperature would typically fall as you move upwards through the cloud. Understanding the height above the cloud base is crucial for meteorologists and pilots who need to know atmospheric conditions. Applying this height in the calculation helps predict weather patterns and ensure aviation safety.